Geodesic
Denominator vectors and compatibility degrees in cluster algebras of finite type
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra, and provide simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero. 
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     author = {Ceballos, Cesar and Pilaud, Vincent},
     title = {Denominator vectors and compatibility degrees in cluster algebras of finite type},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.12795},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12795/}
}
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Ceballos, Cesar; Pilaud, Vincent. Denominator vectors and compatibility degrees in cluster algebras of finite type. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.12795. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12795/

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